Objectives: The living world is embedded in the physical world. Small organisms live in a world of diffusion. The rest of us require a system of ducts for transport of all the materials we need to live. Systems of branched ducts are found in lung, kidney, mammary gland, and many other organs. This project's primary objective is to understand the mechanisms of tissue dynamics that create branched systems. It also aims to clarify the best ways to work with continuum mechanical models of morphogenesis to realistically describe mechanics and transport in developing tissues. Additionally, the project will develop new numerical methods for mixture models with interfaces. We will then test these models in a real time living branching system, the early embryonic lung. Aim 1: Develop accurate, stable, efficient methods for solving mixture problems with sharp interfaces. Aim 2: Develop a suite of models of the mechanical aspects of branching epithelia.